Keçeci Numbers and the Keçeci Prime Number: A Potential Number Theoretic Exploratory Tool

Keçeci Numbers and the Keçeci Prime Number: A Potential Number Theoretic Exploratory Tool Mehmet Keçeci1 1ORCID : https://orcid.org/0000-0001-9937-9839, Türkiye Received: 11.05.2025 Abstract: Keçeci Numbers (first defined: July 27, 2022) constitute a unique numerical system that generates sequences based on specific initial conditions and a set of iterative rules. This system relies on a complex interplay of fundamental arithmetic operations (addition), special divisibility rules (periodically dividing by 2 or 3), and the primality properties of numbers. One of the most notable aspects of Keçeci Numbers is their definability across various number sets, such as integers, rational numbers, negative numbers, complex numbers, and quaternions, exhibiting distinct behaviours in each. This offers a rich ground for investigating how fundamental number theoretic principles manifest within different algebraic structures. The "Keçeci Prime Number" refers to the number (or its positive integer representative, based on the absolute value of its real or scalar part, according to the number type) that is prime and appears most frequently within the generated Keçeci Number sequence. This value can be interpreted as a kind of "characteristic signature" or "condensation point" regarding the sequence's structure. The potential significance of Keçeci Numbers lies not in being an established sequence, but in providing a "computational laboratory" for exploring how fundamental concepts in number theory (divisibility, primality, arithmetic progression) can generate new and unpredictable patterns under specific constraints and conditions. The application of such rules, particularly in higher-dimensional number systems like complex numbers and quaternions, might produce sequences exhibiting fractal-like structures or chaotic behaviour. Such sequences could potentially inspire models for certain dynamic systems in theoretical physics, or new designs for pseudo-random number generators in signal processing engineering or cryptography. As Keçeci Numbers are still in an exploratory phase, their true value will emerge with deeper mathematical analysis in the future (e.g., concerning the long-term behaviour of the sequences, their limits, periodicity, or the distribution of Keçeci Prime Numbers). Such original numerical systems demonstrate that mathematics is not only about solving existing problems but is also a self-enriching discipline that generates new structures and questions. This can serve as a source of motivation, reminding students and researchers of the vast, unexplored territories within the infinite world of numbers. Keywords: Number Sequences, Iterative Processes, Complex Numbers, Rational Numbers, Quaternions, Algorithmic Number Generation, Keçeci Numbers, KececiNumbers, Keçeci Prime Number